The Cheesegrater: Richard Rogers sprinkles the Square Mile

Richard Rogers and his team have conquered the City of London skyline – with an elegant new tower that takes his hi-tech, inside-out architecture to new heights. Oliver Wainwright catches the view from the top of 'the Cheesegrater'

Menagerie of silhouettes … the Leadenhall Building, to the right of the Gherkin, on the City skyline. Photograph: John Safa

'I can claim the credit for this one," says Peter Rees, chief planning officer of the City of London, with a laugh. "When I first saw a model of the building, I told Richard Rogers I could imagine his wife, Ruthie, using it to grate parmesan. I don't think he was too happy, but it stuck."

Rees is talking about the Leadenhall Building, a 225m-tall skeletal glass-and-steel wedge that soars up from the streets of the City, the latest arrival in an ever-growing menagerie of novelty silhouettes that he has ushered in during his 28 years as choreographer of this coveted stretch of skyline. Designed by Rogers Stirk Harbour and Partners, the new colossus's nickname has already caught on: the Cheesegrater.

Buildings with curious monikers are, however, nothing new to the Square Mile. Centuries ago, the Church of St Andrew on Leadenhall Street was given the nickname Undershaft, as it stood in the shadow of a giant maypole that was eventually destroyed by Puritans in the 1540s. Today, with a shaft of a very different kind now looming over the church, buildings in the City no longer draw their names from centuries-old rituals, but from what they look like if you squint a bit.

The steep, triangular mass of the Cheesegrater has now reached its full height, an event that will be celebrated in a topping-out ceremony tomorrow, even if the building won't be fully finished until next year. Alongside it stands the curvaceous Gherkin and, to the south and bulging like a bouncer, the Walkie-Talkie. In addition to these towers, there are plans for a building shaped like a can of ham, another like a knife, and yet another like a rolled-up napkin. The City of London is becoming a bizarre dinner party in the sky.

"Nicknaming is important," says Rees. "The fact that the public warmed to tall buildings after the Gherkin is indicated by the fact that they wanted names for them: they wanted distinctive buildings they could relate to."

While its name is the product of a planning-meeting quip, the Cheesegrater's form is dictated by a building that has been the most powerful force in shaping London for the last three centuries. The dome of St Paul's Cathedral, almost a mile to the west, is protected by a convoluted array of height limits and viewing corridor rules. All buildings must bow to these. This perverse favouritism has produced some disasters – such as Jean Nouvel's One New Change shopping centre. Its attempt to limbo under the radar resulted in a clumsy hulk. But with the Cheesegrater, Rogers and his team have used this strange doctrine of deference to great effect, crafting a slender tapering form of rare elegance. The fact that the tower tilts back to the north isn't just to add some visual dynamism – it's also, like an awkward relative leaning out of a wedding photograph, making sure it doesn't spoil the backdrop of the hallowed dome when seen from Fleet Street.

This constraint has also had a beneficial commercial side effect. As the tower rises, the floors diminish, from a rambling 20,000 sq ft at the lower levels to boutique 6,000 sq ft penthouses, aimed at the cream of asset management firms. While the Walkie-Talkie swells out to maximise profits at the higher levels, Leadenhall offers a more diverse range of spaces. This has proved irresistible to insurance giants Aon and Amlin: half the building will be let to them.

At ground level, the sloping form is also a piece of deference, a way for the building to respect its neighbours' space. And these are no ordinary neighbours: directly across the street stands Rogers' Grade I-listed Lloyd's building, a symphony of ducts and pipes that partner Graham Stirk worked on 30 years ago when he first joined the practice, fresh out of college. "Let's say it's been quite an intimidating context to work in," he grins, pointing out how the new tower is also sandwiched between a listed Edwin Lutyens building and two ancient churches.

The radical response to this precious, tightly packed spot, so typical of the narrow City streets, has been to do away with the lower floors altogether. The entire 48-storey pile has been jacked up a whopping 30 metres into the air, improbably supported by zig-zagging steel legs that stride around the perimeter. The cavernous void beneath, conceived of as an extension of the adjacent square, will be planted with all the trappings of a public space: lawns, trees, cafes, benches and paved areas that will be "activated" by a programme of events. The space will be controlled by Broadgate Estates, the company that blocked the Occupy protesters from Paternoster Square, by St Paul's, in October 2011. It joins the many pseudo-public spaces (ie, actually private) that the firm runs, from Liverpool One to More London.

While the nature of any contemporary public space is becoming increasingly questionable, this is by far the most generous contribution made by any recent building in the Square Mile. It promises to be a dramatic place, boasting a vertical ballet of vast escalators shuttling people up to a suspended lobby and cantilevered restaurants beyond. This seven-storey space was made possible by the tower's unusual structure, laid out on the facade in typical Rogers fashion. A super-sized steel lattice wraps around the building, creating a "diagrid" of flying beams that intersect at star-shaped joints six metres tall – each weighing 30 tonnes and fixed with chunky "mega-bolts". It's a bit like standing in the middle of some giant Meccano creation, right down to the vibrant colour palette. "This is like bridge engineering," says project architect Andy Young, explaining how the entire building was built off-site, then craned into place bit by bit. As a result, only 150–200 workers were needed, compared with the 500–600 usually required for a project of this scale.

Recalling Lloyd's and the Pompidou before it, the Cheesegrater's building-as-machine ethos is celebrated everywhere you look, with each component exposed and painted in jovial colours – just like the cogs of Stirk's transparent watch. On its north side, 22 glass lifts, strapped into orange frames, will hurtle passengers up the yellow shafts at eight metres per second, driven by bright green motors at the summit. Even the building's shock-absorbing springs have been exposed. "People sit and watch the lifts run up our other buildings," says Stirk, barely able to contain his excitement at these new capsules. "But this will be something else." Naturally, the toilets will be visible, too, or partly, their red-and-blue glazing forming part of the outer wall. It all adds up to a giddying urban cliff of lifts and loos. From the top, I had to crane my neck to make out the Lloyd's building, a mere speck way below.

The Cheesegrater seems to have already caught the popular imagination as a spirited building, a cheerful echo of the tapering Shard. After such recent RSH+P projects as London's Neo Bankside and One Hyde Park – that vacant silo for sheiks in Kensington where "public gardens" cower behind metal bars and bulletproof glazing – it has been easy to lose faith in the public-minded rhetoric of Rogers's earlier career. While the Leadenhall is of course a silo for another kind of City elite, it is an intelligent response to the necessities of its site, one that attempts to give as much back to the City as it can. Let's just hope it's not upstaged by all the dinner-party guests still to come.